Constant phase distributed impedance

ABSTRACT

An electrical device is described herein which has an input impedance of substantially constant phase angle over a limited frequency range. It consists of a distributed circuit having a capacitive and resistive medium between two electrodes, of appropriate properties. Implementations can use thin film techniques or monolythic techniques.

United States Patent [1 1 Nathan et al.

[4 1 Feb. 25, 1975 CONSTANT PHASE DISTRIBUTED IMPEDANCE [76] Inventors:Amos Nathan, Haifa, Israel; Reed K.

Even, Middletown, NJ.

[22] Filed: Feb. 22, 1972 [21] Appl. No.: 227,715

[30] Foreign Application Priority Data Oct. 19, 1971 Great Britain48502/71 [52] U.S. Cl 330/38 M, 330/185, 333/70 CR [51] Int. Cl. 1103f3/14 [58] Field of Search 333/70 CR, 70 S, 21 R,

[56] References Cited UNITED STATES PATENTS 1/1949 Boghosian et al.333/70 CR X 7/1965 Barditch et al 333/70 CR 3,233,196 2/1966 Osafune etal. 333/70 CR 3,371,295 2/1968 Bourgault et al 333/70 CR 3,432,7783/1969 Ertel 333/70 CR FOREIGN PATENTS OR APPLICATIONS 1,349,378 12/1963France 330/70 S OTHER PUBLICATIONS Electronics, Sept. 4, 1959, pp.44-49, Vol. 32 No. 36, Network Design of Microcircuits, by Hagar.

Primary Examiner-Nathan Kaufman [57] ABSTRACT An electrical device isdescribed herein which has an input impedance of substantially constantphase angle over a limited frequency range. It consists of a distributedcircuit having a capacitive and resistive medium between two electrodes,of appropriate properties. 1mplementations can use thin film techniquesor monolythic techniques.

7 Claims, 11 Drawing Figures PPJENTEU FEB 2 51975 sum 1 UF 2 CONSTANTPHASE DISTRIBUTED IMPEDANCE This invention relates to distributed RCstructures whose input impedance has a substantially frequencyindependent, i.e. constant, phase angle over a range of frequencies.

The phase angle is defined as the angle between electrical current andvoltage in the steady state sinusoidal mode of operation, or,equivalently, as the angle between them if they are expressed as complexquantities, as well known in the description of alternating currents.Impedance is defined as the ratio of complex voltage and complexcurrent. R stands for resistance and C denotes capacitance.

It is well known in the art how to implement impedances of substantiallyconstant phase angle over some frequency range in lumped electricalcircuits. Such impedances will, for short, be called in the sequelconstant phase impedance." For example, in the following paper: R.Morrison, RC constant argument driving point admittances," IRETransactions of Circuit Theory, volume CT-6, pages 310-317, September,1959, RC ladder circuits are described for this purpose.

This invention provides constant phase impedances in devices usingdistributed structures.

It is thus an object of the invention to provide constant phasedistributed impedance.

It is a further object of the invention to provide constant phaseimpedance operating over a wider frequency range or with a better phaseconstancy than prior art lumped circuits.

Other objects of the invention are implementations of constant phaseimpedance in thin film and in monolythic devices.

The invention possesses other objects and features of advantage, some ofwhich of the foregoing will be set forth in the following description ofthe preferred form of the invention which is illustrated in the drawingsaccompanying and forming part of this specification. It is to beunderstood, however, that variations in the showing made by the saiddrawings and description may be adopted within the scope of theinvention as set forth in the claims.

FIG. 1 is a perspective view of an embodiment of a device according tothe invention;

FIG. 2 is a sectional cut through such a device and also defines aconceptual system of coordinates;

FIG. 3 is a schematic diagram of an equivalent circuit for a smallsection of the device;

FIG. 4 is a section through the device of FIG. 1 and also includes aschematic diagram of a compensating lumped RC network;

FIG. 5 is a section through another embodiment of the inventionaccording to FIG. 4;

FIGS. 6 and 7 are sectional cuts through yet further embodiments of theinvention;

FIGS. 8, 9, and 10 are schematic diagrams showing the use of theinvention for the provision of constant phase transfer function devices;

FIG. 11 is a curve typical of phase deviations from the desired constantvalue, in devices according to the invention.

The theory and practice of devices according to the invention will nowbe described. v

In FIGS. 1 and 2, l and 2 are conductive electrodes, Q is a dielectriclayer, and Q a resistive layer. The device is accessible throughterminals shown in FIG. 2, which are conductively connected toelectrodes 1 and g. The constant phase impedance is produced acrossterminals 5 xyz is a Cartesian system of coordinates. The section of thedevice between coordinates x and .r dx is equivalent to the seriescombination of capacitance c(x)d.r and conductance g(x)dx, as showndiagramatically in FIG. 3. Conductance g(.r) is the reciprocal ofresistance r(x), i.e., g(x) l/r(.r). In this instance, conductance,capacitance, and resistance are to be understood as specific values,i.e. per unit length." The admittance of such a section is then given byH y(j g( /I g( )/(j where cu is the angular frequency, defined as 2 11-times frequency.

If the structure extends from x x, to x x (in the x direction) then theadmittance between terminals is givenby n 22 YUw)=f y(jw,x)dx

Admittance is defined as the reciprocal of impedance. Defining where 1(x) is an arbitrary differentiable function and (x) is the inversefunction, defined through then the admittance function m, U 1( )l n(corresponds to a structure extending in the x direction from x (x to x(x that likewise realizes the admittance Y (j w). The transition from yto y'through use of 17(x) will be called a scale transformation. Thus,any realization c rresponding to some function y (j w, x) isequivalent/to a family of realizations generated from it by all possiblescale transformationsj here denotes the square root of l.

If the structure is infinite (in the x direction) then a constant phasedistributed structure results from the laws:

where R and C are constant resistance and capacitance, respectively, anda and b are constants. Defining k a/b the constant phase angle is e=l/l+k'1r/2=1r/2 b/a-i-b andthe inputadrnittance can be shown to be givenby As remarked above, it is still possible to perform scaletransformations and thereby obtain equivalent realizations.

The relations are transformed through a scale transformation into 3 Ifthe device were infinite, then, with the relations c(x) C, if; 'y(x) R,e

- and the definitions there follows, as can be shown,

where L L are finite, and the corresponding structure extends from x L,to x L in the x-direction. Two specific examples will now be given, eachof length 2L, with the structure extending from L to +L. (a) With 17(x)=L sgn X In (I IxI/L; (x)= L sgnx there follows (b) With I n(x) 2/90 Ltan (/n'x/L); (x) 2/90 L tan /mx/L) I there follows 7 FIGTGTot to scalei1s a gerieial view of a device embodying one implementation of theinvention and FIG. 4 is the corresponding sectional view. FIG. 2 is asection through a more general embodiment of the invention and alsodefines an associated conceptual coordinate system. In the deviceaccording to FIG. 4, the required laws for C(x) and r(x) are produced byhaving variable width of structure, w(x), in the y direction andvariable height, h(x), of the resistive layer, in the z direction. Adielectric layer, 2, of dielectric constant e and constant thickness dhas variable width w(x) and is adjacent to electrode 1. Resistive layerf thickness h(x) fill the space between the said dielectric layer and asecond electrode 2. The part of the structure between x x, and x xsatisfies the laws found above for r(x) and C(x), provided that, in thisinterval,

In the embodiment of FIG. 4, in parallel with this structure, there areconnected between electrodes 1 and 2 capacitor 9 and resistor 10 for theimprovement of the constancy of phase angle. Alternatively, resistor andcapacitor can be replaced by distributed structures. For example, asshown in FIG. 5, the required capacitance extends from x to X1 and isshown at 7, and the required resistance extends from x to x, and isshown at 8. In this example, the latter is in series with a largecapacitance formed through dielectric medium 6, in order to provideinfinite impedance at zero frequency, which is at times of advantage.The width and height of the structure in the regions x to x, and x to x,are constant, in this example. The effect on the input impedance of theparts of structure from x to x, and from x to x, is to compensate forthe cut off infinite lengths at both ends of the variable resistancevariable capacitance part of the structure extending from x, and xrather than from minus to puls infinity. If properly chosen, theseadditions improve performance.

Denoting the dielectric constant of layer 3 in FIGS. 2, 3, and 4 by eand the resistivity of layer 4 by p there holds By use of a scaletransformation, this structure can be transformed into another one ofconstant width. For this purpose we take the function 7( In m l- (4 2;where ,u is a constant, and obtain for c(x) and r(x) the relations 0 7(0 w and for the end points of the shaped structure (formerly x to xrespectively.

Similarly, a scale transformation with (v 0; p. a) /2vx 1) x ,u. e'" /2where u, v are constants, transforms C(X) C eax into 0 (ILX which is alinear function of x.

Note also that 11(x) x corresponds to r(x) r(x), y( y( For a deviceaccording to the invention and extending from x x, to x x and such thatlet the frequency range of operation be defined by L A a These relationspermit the easy approximate calculation of the required values of thecompensating lumped capacitance and resistance. y 'y must be variedaround unity in order to obtain the best results for givenspecifications.

A specific example of the invention embodying the principles anddescriptions as hereinbefore set forth is as follows: For a phase angleof (b =45 and a length of structure corresponding to -ax ax 3, with 'y,1.9 and 7 1.7; and operating range for w of between 0.07 (n and 10.0there resulted a phase angle error of the order of not more than 1.

In one embodiment of the invention according to FIG. 4 the resistivelayer is composed of cermet with a resistivity of the order of p 1 ohmmeter and the capacitance is composed of tantalum oxide with acapacitance per unit area of the order of c 10 ,uF/m Taking 4) 45 whichcorresponds to 1r/4 radians, it follows that a b and p 0. We take x, l0meters and x meters, so that the length of the structure is x -x 2centimeters and a=b= l l5/meter. In this example, the maximum value of his 10 meters and its minimum value is 10 meters. Electrodes l and 2 arethemselves made of tantalum and the capacitive layer is produced byoxidation. W, 6.3 l0" meters, so that w(x) increases from a minimumvalue of 2X10 meters to a maximum value of 2X10 meters. h,, 10' meters.The device performs properly around an angular frequency of the order ofor 1.60 Megahertz, approximately. The compensating capacitor andresistor have the approximate values c. E 1.73 p. F; R E 0.058 ohms,

respectively. In another embodiment the compensating capacitor andresistor are as above, but the compensating resistor is placed serieswith a capacitor having a capacitance that presents a virtual shortcircuit at the operating frequencies; i.e., it is sufficiently great.

A further embodiment of the invention corresponding to FIG. 4 uses asilicon wafer doped on one side so as to form electrode 2, the doping onthis side being such as to provide good conductivity. The rest of thewafer, with the exception of the opposite face, has constant doping andvariable height and implements the required law of resistance, and theother face is uniformly oxidized, forming silicon oxide. 1 is anadjacent metal electrode, and the capacitance is formed between it andthe bulk of the wafer through a layer of silicon oxide. Typical valuesare: Resistivity of the silicon resistor: p 10 to 10 am, MOS capacitor:0 3 l0 ai /m The capacitor formed according to the above description iscalled an M08 capacitor. Maximum height of the resistive layer: 2.5 10"meters. The middle angular frequency w is approximately 4 l0 secor about60 kHz. lf gallium arsenite is used instead of Silicon as basicmaterial, the middle frequency is about 1 kHz, for a ratio of maximum tominimum height of the variable height part of the device of 100 to I.

It is also possible to use a layer of constant height and variabledoping in order to implement the required law of resistance. Variableheight and doping can also be used for the formation of the requiredresistance function.

FIGS. 6 and 7 relate to a yet further implementation of the invention.12 is an insulating substrate, such as glass, for example. Theelectrodes are 1 and 2. Electrode 1 is oxidized to form a dielectriclayer 3 of constant thickness. Electrode 2 is oxidized in one example.In another example it is not oxidized. The electrodes are made, in oneexample of the invention, of tantalum and the dielectric consists oftantalum oxide. The structure is covered by a resistive layer 4. Theresistance law between the electrodes is controlled by variations in thedistance between them. These variations are not shown in the FIGURES.The comb like arrangement in FIG. 7 merely provides a convenient methodto accommodate a great length of structure in a limited space, or with alimited length. If the structure of FIG. 6 is used, then FIG. 7corresponds to a view from above of the section shown in FIG. 6. In theexample of FIG. 7 the length of structure is only somewhat greater thanone sixth its effective length, because there are exactly six slotsbetween the teeth of the comb. Layer 4 is not shown in FIG. 7, nor arethe variations in slot width in dicated.

In FIGS. 8, 9, and 10 it is shown how the invention can be used for theprovision of constant phase transfer functions. In FIG. 8, l5 and 16 areinput and output terminals, respectively. 19 is a high gain operationalamplifier, l3 is the input impedance and 17 is the feedback impedance.In this example, 17 is aresistor and 13 is a constant phase impedanceaccording to the invention. The device provides the substantiallyconstant phase transfer function between terminals 15 and 16. In thedevice corresponding to FIG. 9 the input impedance is provided byresistor 20 and the feedback im pedance is provided by a constant phaseimpedance 14 according to the invention. In FIG. 10 both input impedancel3 and feedback impedance 18 are constant phase impedances according tothe invention, corresponding, however, to different phase angles. Inthis latter example the operating ranges of impedances l3 and 18 must besubstantially the same and the phase angle between input terminal 15 andoutput terminal 16 is given by the difference between the phase anglesof 13 and 18.

It is also possible to replace resistors 17 and 20 by differentimpedances. For example, replacing resistor 17 by a capacitor, thetransfer phase angle is shifted by ninety degrees.

FIG. 11 is a typical performance curve of an imped ance according to theinvention. The curve corresponds to a device for a phase shift of 9. andalso for a phase shift of 81, the two curves being co-incident. Thecurves are obtained for such a device if compensating resistors andcapacitors are used as hereinbefore specified with values ofcoefficients y, Y2 l. The scale of abscissae is logarithmic. Theordinates are the deviation of the actual phase from the desired phasedz.

What we claim is:

1. An electrical device having an input impedance of substantiallyconstant phase over a finite range of frequencies comprising:

impedance means having a first and a second surface,

the maximal distance therebetween being small compared with the laterdimensions thereof;

first and second electrically conductive means substantially coveringand contiguous to said first and second surfaces, respectively;

one of said lateral dimensions defining length;

where (the law of) said impedance means is constituted so that itsimpedance per unit length (of said impedance means) is substantiallygiven by e is the basis of natural logarithms, x is a coordinatemeasuring said length, and R are predetermined non-vanishing constantcapacitance and resistance, respectively, a and b are predeterminedconstants such that their product does not vanish, 17(x) is apredetermined non-constant function of x, d/dx 1 (x) is the derivativeof 1;(x) with respect to x,j is the square root of l, and w is'theangular frequency;

said input impedance being the impedance between said plural conductivemeans.

2. The device as recited in claim 1 wherein said impedance means iscomprised of dielectric means having said first surface and resistivemeans (adjacent to) contiguous with said dielectric means and havingsaid second surface and so constitrited that the laws of capacitanceC(x) and resistance r(x) per unit length of said dielectric means andsaid resistive means are substantially determined by respectively.

3. The device as recited in claim 2 wherein "1 (x) x and c(x) and r(x)are substantially given by exponential relations as functions of saidcoordinate x.

4. The device as recited in claim (5) 2 wherein said dielectric (layer)means has substantially constant height d and the required law ofcapacitance C(X) is implemented through variations with x of width w(x)(thereof) of said impedance means (and) substantially according to therelation wherein e is a constant and wherein the width of said layer islikewise w(x)) and the required law of resistance r(x) is implementedthrough variations with x of (its) height h'(x) of said resistive meanssubstantially according to the relation wherein p is a constant; heightbeing measured perpendicularly to said lateral dimensions and widthbeing defined as the lateral extension perpendicular to said lengthdimension.

5. The device as recited in claim 2 wherein said plural conductive meansare comprised of metal electrodes, and said dielectric means iscomprised of a layer of metal oxide oxidized onto one face of one ofsaid electrodes.

6. The device as claimed in claim 2 (and including) wherein said deviceis composed of a Silicon wafer having a first and a second face anddoped on one of said faces to provide a low-resistance electrode, saidelectrode forming said first conductive means; and second face of saidwafer oxidized to provide said dielectric (layer) means; a metalelectrode (placed adjacent) contiguous (thereto) therewith to providesaid second conductive means; (a second doped layer) the material ofsaid wafer between said first electrode and said dielectric (layer)means doped to provide said resistive means.

7. The device as recited in claim 1 and including a lumped electricalnetwork having first and second connections, connected (at one end)thereat to said first (conductive means and connected at a second end tosaid) and second conductive means, respectively, for

extending said frequency range.

1. An electrical device having an input impedance of substantially constant phase over a finite range of frequencies comprising: impedance means having a first and a second surface, the maximal distance therebetween being small compared with the later dimensions thereof; first and second electrically conductive means substantially covering and contiguous to said first and second surfaces, respectively; one of said lateral dimensions defining length; where (the law of) said impedance means is constituted so that its impedance per unit length (of said impedance means) is substantially given by z(x) (Roe b (x) + 1/j omega Co ea (x))/(d/dx eta (x) ), e is the basis of natural logarithms, x is a coordinate measuring said length, Co and Ro are predetermined non-vanishing constant capacitance and resistance, respectively, a and b are predetermined constants such that their product does not vanish, eta (x) is a predetermined non-constant function of x, d/dx eta (x) is the derivative of eta (x) with respect to x, j is the square root of -1, and omega is the angular frequency; said input impedance being the impedance between said plural conductive means.
 2. The device as recited in claim 1 wherein said impedance means is comprised of dielectric means having said first surface and resistive means (adjacent to) contiguous with said dielectric means and having said second surface and so constitrited that the laws of capacitance c(x) and resistance r(x) per unit length of said dielectric means and said resistive means are substantially determined by c(x) Co e a (x) d kappa (x)/dx gamma (x) Ro e b (x) / (d eta (x)/dx ) respectively.
 3. The device as recited in claim 2 wherein eta (x) x and (c(x) Co e ax ) gamma (x) Ro e bx ) c(x) and r(x) are substantially given by exponential relations as functions of said coordinate x.
 4. The device as recited in claim (5) 2 wherein said dielectric (layer) means has substantially constant height d and the required law of capacitance c(x) is implemented through variations with x of width w(x) (thereof) of said impedance means (and) substantially according to the relation w(x) 1/ epsilon d. c(x) wherein epsilon is a constant ( and wherein the width of said layer is likewise w(x)) and the required law of resistance r(x) is implemented through variations with x of (its) height h(x) of said resistive means substantially according to the relation h(x) 1/ Rho r(x). w(x) wherein Rho is a constant; height being measured perpendicularly to said lateral dimensions and width being defined as the lateral extension perpendicular to said length dimension.
 5. The device as recited in claim 2 wherein said plural conductive means are comprised of metal electrodes, and said dielectric means is comprised of a layer of metal oxide oxidized onto one face of one of said electrodes.
 6. The device as claimed in claim 2 (and including) wherein said device is composed of a Silicon wafer having a first and a second face and doped on one of said faces to provide a low-resistance electrode, said electrode forming said first conductive means; and second face of said wafer oxidized to provide said dielectric (layer) means; a metal electrode (placed adjacent) contiguous (thereto) therewith to provide said second conductive means; (a second doped layer) the material of said wafer between said first electrode and said dielectric (layer) means doped to provide said resistive means.
 7. The device as recited in claim 1 and including a lumped electrical network having first and second connections, connected (at one end) thereat to said first (conductive means and connected at a second end to said) and second conductive means, respectively, for extending said frequency range. 